7 problems

Positivity conjecture. The functional $\omega_\beta$, normalized by

The Connes–Marcolli system has KMS$_1$-states corresponding to measures on the relevant space satisfying the scaling condition. In particular, measures supported on the full-measur…

Let $({\mathfrak M},\tau)$ be a $W^\ast$-dynamical system on a Hilbert space ${\mathcal H}$, and let $\omega(A)=(\Omega,A\Omega)$ be a faithful $(\tau,\beta)$-KMS vector state on…

Uniqueness conjecture. The state $\omega_{\mathbb{Z}^D}^\beta$ obtained in this way is the only thermodynamic-limit $\beta$-KMS state.

Phase-transition conjecture. For $n>2$, a phase transition occurs at

Let the $GSp_{4}$-system be the dynamical system under consideration, and let a $\textmd{KMS}_\beta$ state mean a KMS state at inverse temperature $\beta$. The two cases in questio…

Let $H$ be the potential described above, with $\beta=1$, and let ${\cal U}(\mu)$ be the associated system. Let $\nu_1$ be the unique eigenmeasure for ${\cal L}_{\beta=1}^*$, and l…